The aim is to study the statistical problems of probability estimation, and related significance tests for categorical and continuous distributions and for distributions of mixed type, categorical in some dimensions and continuous in others, and to apply the theory to real data especially medical because of its potential importance in medical diagnosis. This research leads rapidly to problems concerning the foundations of statistics. Methods that make use of personal probability judgments are called Bayesian, and those that seem not to do so are called non-Bayesian. The project will continue to make use of both kinds of methods and especially a synthesis of them called the Bayes/non-Bayes compromise, as well as a technique called hierarchical Bayes. This technique is especially relevant for significance tests for categorical data when mixed Dirichlet priors are used. Another technique of value in probability density estimation is known as maximum penalized likelihood in which a roughness penalty is introduced. This is required because the familiar method of maximum likelihood, when applied to nonparametric density estimation, leads to an infinitely rough density curve. Attempts will be made to expound the methods in a connected whole, and to publish programs, so as to enable other statisticians to understand, appreciate, and use the methods when they are fully developed. Part of the plan is to compare these methods with those proposed by other workers. The project requires a mixture of the techniques and philosophy of statistics as well as mathematical methods and computation.